On Cohomology Vanishing with Polynomial Growth on Complex Manifolds with Pseudoconvex Boundary

The partial derivative cohomology groups with polynomial growth H-p,g.(r,s) that, given a complex manifold M, a locally pseudoconvex bounded domain Omega (sic) M satisfying certain geometric boundary conditions and a holomorphic vector bundle E -> M, H-p.g.(r,s) (Omega, E) = 0 holds for all s >= 1 if E is Nakano positive and r = dim M. It will also Hp.g.(r,s)(Omega, E) = 0 for all r and s with r+s > dim M if, moreover, rank E = 1. By the comparison theorem due to Deligne, Maltsiniotis (Asterisque 17 (1974), 141-160) and Sasakura (Inst. Math. Sci. 17 (1981), 371-552), it follows in particular that, for any smooth projective variety X, for any ample line bundle L -> X and for any effective divisor D on X such that [D vertical bar(vertical bar|D vertical bar) >= 0, the algebraic cohomology H-alg(s)(X \ vertical bar D, Omega(r)(X) (L)) vanishes if r + s > dim X.